This question is somewhat related to my previous one on [Grassmanians.][1] The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very theoretical setting: in the right setting, it exists, it is unique (if the setting is *really* nice), and you can integrate against it to define new objects that will have nice properties because the measure itself does.

So my question is: how practical is it to compute with? I'm talking about very concrete examples here, e.g. "G=O(4), I integrate f(M)=[some explicit function with a matrix input] and the answer I get is 42".
From conversations, I got the feeling that the construction of the Haar measure allows you *in principle* to write such a computation explicitly. I'm concerned with the tractability of the computation itself. Examples would be great.

  [1]: http://mathoverflow.net/questions/35655/measure-on-real-grassmannians